task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | ## Task B-4.4.
In some numerical system, with a base less than 25, the product of a two-digit number with identical digits and its double is equal to 1210 (in the same numerical system). What is the number and in which numerical system? | 22_8 | 60 | 4 |
math | Among the points corresponding to number $1,2,...,2n$ on the real line, $n$ are colored in blue and $n$ in red. Let $a_1,a_2,...,a_n$ be the blue points and $b_1,b_2,...,b_n$ be the red points. Prove that the sum $\mid a_1-b_1\mid+...+\mid a_n-b_n\mid$ does not depend on coloring , and compute its value. :roll: | n^2 | 109 | 3 |
math | 1. Determine the functions $f: N^{*} \rightarrow N^{*}$ that satisfy the relation $\frac{f^{2}(n)}{n+f(n)}+\frac{n}{f(n)}=\frac{1+f(n+1)}{2},(\forall) n \geq 1$. Cătălin Cristea, Craiova (GMB 9/2013) | f(n)=n,\foralln\inN^{*} | 87 | 13 |
math | 14. Let \( f(x) = \frac{a^x}{1 + a^x} \) where \( a > 0, a \neq 1 \), and \([m]\) denotes the greatest integer not exceeding the real number \( m \). Find the range of the function \(\left[f(x) - \frac{1}{2}\right] + \left[f(-x) - \frac{1}{2}\right]\). | -1or0 | 98 | 4 |
math | Example 6 Consider a $2^{m} \times 2^{m}\left(m \in \mathbf{Z}_{+}\right)$ chessboard, which is divided into several rectangles composed of the squares of the chessboard, and each of the $2^{m}$ squares on one of the diagonals is a rectangle of side length 1. Find the minimum sum of the perimeters of the divided rectan... | 2^{m+2}(m+1) | 102 | 10 |
math | What is the largest integer that cannot be expressed as the sum of 100 not necessarily distinct composite numbers? (A composite number is one that can be expressed as the product of two integers greater than 1.) | 403 | 44 | 3 |
math | 1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=3000$, if $f(0)=1$ and for any $x$ the equality $f(x+2)=f(x)+3 x+2$ holds. | 6748501 | 61 | 7 |
math | $$
x^{3}+a x^{2}+b x+6=0
$$
In the equation, determine $a$ and $b$ such that one root is 2 and another root is 3. What is the third root? | x_{3}=-1 | 54 | 6 |
math | Example 6 Given that $x, y, z$ are real numbers, and $x+y+z=5, xy+yz+zx=3$. Try to find the maximum and minimum values of $z$.
(Canadian 10th High School Mathematics Competition) | -1 \leqslant z \leqslant \frac{13}{3} | 57 | 21 |
math | Find all natural numbers $n$ for which the following $5$ conditions hold:
$(1)$ $n$ is not divisible by any perfect square bigger than $1$.
$(2)$ $n$ has exactly one prime divisor of the form $4k+3$, $k\in \mathbb{N}_0$.
$(3)$ Denote by $S(n)$ the sum of digits of $n$ and $d(n)$ as the number of positive divisors of $n... | 222 \text{ and } 2022 | 150 | 13 |
math | 1. (40 points) Let real numbers $a_{1}, a_{2}, \cdots, a_{n} \in[0,2]$, and define $a_{n+1}=a_{1}$, find the maximum value of $\frac{\sum_{i=1}^{n} a_{1}^{2} a_{i+1}+8 n}{\sum_{i=1}^{n} a_{i}^{2}}$. | 4 | 101 | 1 |
math | 5. Find all values of the parameter $a$ for which the equation $\left(\left(1-x^{2}\right)^{2}+2 a^{2}+5 a\right)^{7}-\left((3 a+2)\left(1-x^{2}\right)+3\right)^{7}=5-2 a-(3 a+2) x^{2}-2 a^{2}-\left(1-x^{2}\right)^{2}$ has two distinct solutions on the interval $\left[-\frac{\sqrt{6}}{2} ; \sqrt{2}\right]$. Specify the... | \in[0.25;1),x_{1}=\sqrt{2-2},x_{2}=-\sqrt{2-2}\\\in[-3.5;-2),x_{1}=\sqrt{--2},x_{2}=-\sqrt{--2} | 139 | 63 |
math | ## 12. Beautiful Lace
Two lacemakers need to weave a piece of lace. The first one would weave it alone in 8 days, and the second one in 13 days. How much time will they need for this work if they work together? | 4.95 | 56 | 4 |
math | The sides of triangle are $x$, $2x+1$ and $x+2$ for some positive rational $x$. Angle of triangle is $60$ degree. Find perimeter | 9 | 39 | 1 |
math | 12. (2001 National High School Competition Question) The range of the function $y=x+\sqrt{x^{2}-3 x+2}$ is | [1,\frac{3}{2})\cup[2,+\infty) | 34 | 18 |
math | Exercise 3. Consider a number $N$ that is written in the form $30 x 070 y 03$, with $x, y$ being digits between 0 and 9. For which values of $(x, y)$ is the integer $N$ divisible by 37? | (x,y)=(8,1),(4,4),(0,7) | 65 | 15 |
math | 6. Variant 1. Grisha thought of such a set of 10 different natural numbers that their arithmetic mean is 16. What is the maximum possible value of the largest of the numbers he thought of? | 115 | 45 | 3 |
math | Find all triplets $(a, b, c)$ of positive integers, such that $a+bc, b+ac, c+ab$ are primes and all divide $(a^2+1)(b^2+1)(c^2+1)$. | (1, 1, 1) | 55 | 10 |
math | (D) Find the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(f(y)+2 x)=x+f(x)+f(y)
$$ | f(x)=x+ | 42 | 5 |
math | 10. (20 points) Given points $M(-1,0), N(1,0)$, the perimeter of $\triangle M N Q$ is 6, and the trajectory of the moving point $Q$ is the curve $C$. $P$ is any point on the circle $x^{2}+y^{2}=4$ (not on the x-axis), and $P A, P B$ are tangent to the curve $C$ at points $A, B$ respectively. Find the maximum value of t... | \frac{3}{2} | 129 | 7 |
math | 2. Given vectors $\boldsymbol{a} 、 \boldsymbol{b}$ satisfy $|\boldsymbol{a}|=1,|\boldsymbol{b}|=\sqrt{3}$, and $(3 a-2 b) \perp a$. Then the angle between $a 、 b$ is $\qquad$ | \frac{\pi}{6} | 69 | 7 |
math | One. (20 points) Given the parabola $y=a x^{2}+b x+c$ passes through the point $(b,-3)$, and $|a| c+b|c|=0$, and the inequality $a x^{2}+b x+c+3>0$ has no solution. Find all possible values of the triplet $(a, b, c)$.
---
Please note that the formatting and line breaks have been preserved as requested. | \left(-\frac{1}{2}, \frac{1}{2},-\frac{25}{8}\right) | 98 | 27 |
math | Source: 1976 Euclid Part B Problem 2
-----
Given that $x$, $y$, and $2$ are in geometric progression, and that $x^{-1}$, $y^{-1}$, and $9x^{-2}$ are in are in arithmetic progression, then find the numerical value of $xy$. | \frac{27}{2} | 70 | 8 |
math | Sure, here is the translated text:
```
II. (40 points) Find all positive integers $n$, such that for any positive real numbers $a, b, c$ satisfying $a+b+c=1$, we have
$$
a b c\left(a^{n}+b^{n}+c^{n}\right) \leqslant \frac{1}{3^{n+2}}.
$$
``` | n=1, 2 | 92 | 6 |
math | Example 9 Rooster one, worth five coins; hen one, worth three coins; three chicks, worth one coin. A hundred coins buy a hundred chickens. Question: How many roosters and hens are there? | \begin{array}{|c|cccc|}
\hline
x_{1} & 0 & 4 & 8 & 12 \\
\hline
x_{2} & 25 & 18 & 11 & 4 \\
\hline
x_{3} & 75 & 78 & 81 & 84 \\
\hline
\end{array} | 46 | 90 |
math | Consider a polynomial $P(x,y,z)$ in three variables with integer coefficients such that for any real numbers $a,b,c,$ $$P(a,b,c)=0 \Leftrightarrow a=b=c.$$
Find the largest integer $r$ such that for all such polynomials $P(x,y,z)$ and integers $m,n,$ $$m^r\mid P(n,n+m,n+2m).$$
[i]Proposed by Ma Zhao Yu | r = 2 | 94 | 5 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(1+\tan^{2} x\right)^{\frac{1}{\ln \left(1+3 x^{2}\right)}}
$$ | e^{\frac{1}{3}} | 56 | 9 |
math | 15. (6 points) The students of Class 3 (1) are lined up in three rows for morning exercises, with an equal number of students in each row. Xiao Hong is in the middle row. Counting from left to right, she is the 6th; counting from right to left, she is the 7th. The total number of students in the class is $\qquad$. | 36 | 84 | 2 |
math | Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ with centers $(1, 1)$ and $(4, 5)$ and radii $r_1 < r_2$, respectively, are drawn on the coordinate plane. The product of the slopes of the two common external tangents of $\mathcal{C}_1$ and $\mathcal{C}_2$ is $3$. If the value of $(r_2 - r_1)^2$ can be expressed as a co... | 13 | 143 | 2 |
math | A1 (1-2, Romania) Solve the equations in the set of real numbers
(a) $\sqrt{x+\sqrt{2 x-1}}+\sqrt{x-\sqrt{2 x-1}}=\sqrt{2}$;
(b) $\sqrt{x+\sqrt{2 x-1}}+\sqrt{x-\sqrt{2 x-1}}=1$;
(c) $\sqrt{x+\sqrt{2 x-1}}+\sqrt{x-\sqrt{2 x-1}}=2$. | ()\frac{1}{2}\leqslantx\leqslant1;(b) | 104 | 21 |
math | Let $p$ be a prime number, $p \ge 5$, and $k$ be a digit in the $p$-adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$-adic representation. | p - 1 | 64 | 5 |
math | Determine all triples $(x,y,p)$ of positive integers such that $p$ is prime, $p=x^2+1$ and $2p^2=y^2+1$. | (x, y, p) = (2, 7, 5) \text{ is the only solution} | 40 | 24 |
math | 3.42 The digits of a certain three-digit number form a geometric progression. If in this number the digits of the hundreds and units are swapped, the new three-digit number will be 594 less than the desired one. If, however, in the desired number the digit of the hundreds is erased and the digits of the resulting two-d... | 842 | 106 | 3 |
math | 5.1. For the sequence $\left\{a_{n}\right\}$, it is known that $a_{1}=1.5$ and $a_{n}=\frac{1}{n^{2}-1}$ for $n \in \mathbb{N}, n>1$. Are there such values of $n$ for which the sum of the first $n$ terms of this sequence differs from 2.25 by less than 0.01? If yes, find the smallest one. | 100 | 110 | 3 |
math | Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EF GH$ be another regular tetrahedron such that the volume of $EF GH$ is $\tfrac{1}{8}\text{-th}$ the volume of $ABCD$. The height of $EF GH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\tfrac{a}{b}}$, where... | 7 | 118 | 1 |
math | 5. The first term of an arithmetic sequence is 9, and the 8th term is 12. How many terms in the first 2015 terms of this sequence are multiples of 3? | 288 | 45 | 3 |
math | (8) Let $S(n)$ denote the sum of the digits of the positive integer $n$, then $\sum_{n=1}^{2011} S(n)=$ | 28072 | 39 | 5 |
math | ## Task 2
Subtract and justify!
$$
\begin{aligned}
& 66-25=41, \text { because } 41+25=66 \\
& 78-42= \\
& 96-53= \\
& 84-12=
\end{aligned}
$$ | 36,43,72 | 75 | 8 |
math | 1. Find the coefficient of $x^{2}$ in the expansion of $\left(1+x+x^{2}\right)^{9}$. | 45 | 30 | 2 |
math | 8. A line passing through the focus $F$ of the parabola $y^{2}=2 p x$ intersects the parabola at points $A, B$. The projections of points $A, B$ on the directrix of this parabola are $A_{1}, B_{1}$, respectively. Find the size of $\angle A_{1} F B_{1}$. | 90 | 84 | 2 |
math | ## Task B-2.5.
Petar has a set of 9 sports t-shirts. Each t-shirt has one of the numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ printed on it, and all the numbers on the t-shirts are different. Petar is going on a trip and wants to take 5 t-shirts with him, such that at least three of them have an even number printed on... | 45 | 118 | 2 |
math | Three balls of the same radius touch each other pairwise and a certain plane. The base of the cone is located in this plane. All three spheres touch the lateral surface of the cone externally. Find the angle at the vertex of the axial section of the cone, if the height of the cone is equal to the diameter of the ball.
... | 2\operatorname{arctg}\frac{\sqrt{3}}{12} | 68 | 19 |
math | 2.7. Vector $\vec{a}$ forms equal angles with the coordinate axes. Find its coordinates if $|\bar{a}|=\sqrt{3}$. | \vec{}={1,1,1} | 34 | 10 |
math | ## Task B-1.4.
A pedestrian who crosses $1 \mathrm{~km}$ in 12 minutes, will travel the distance from $A$ to $B$ in the same time it takes a cyclist to travel a $10 \mathrm{~km}$ longer route, when the cyclist covers $1 \mathrm{~km}$ in $4 \frac{1}{2}$ minutes. Determine the distance from $A$ to $B$. | 6\mathrm{~} | 96 | 6 |
math | 6.5. In the room, there are 10 people - liars and knights (liars always lie, and knights always tell the truth). The first said: "In this room, there is at least 1 liar." The second said: "In this room, there are at least 2 liars." The third said: "In this room, there are at least 3 liars." And so on,
up to the tenth,... | 5 | 122 | 1 |
math | 22. Let $A$ and $B$ be two positive prime integers such that
$$
\frac{1}{A}-\frac{1}{B}=\frac{192}{2005^{2}-2004^{2}}
$$
Find the value of $B$. | 211 | 65 | 3 |
math | A triangle's sides are: $a=15 \mathrm{~cm}, b=20 \mathrm{~cm}, c=25 \mathrm{~cm}$. What is the length of the angle bisector $A A_{1}$? | 21.08 | 54 | 5 |
math | 4. (8 points) There is a magical tree with 60 fruits on it. On the first day, 1 fruit will fall. Starting from the second day, the number of fruits that fall each day is 1 more than the previous day. However, if the number of fruits on the tree is less than the number that should fall on a certain day, then on that day... | 14 | 119 | 2 |
math | Paris has two million inhabitants. A human being has, at most, 600000 hairs on their head. What is the largest number of Parisians that we can hope to find who have exactly the same number of hairs on their head? | 4 | 52 | 1 |
math | Three, given $x, y \in N$, find the largest $y$ value such that there exists a unique $x$ value satisfying the following inequality:
$$
\frac{9}{17}<\frac{x}{x+y}<\frac{8}{15} \text {. }
$$ | 112 | 63 | 3 |
math | Find all square numbers whose every digit is odd. | 1,9 | 10 | 3 |
math | Find all polynomials $P\in \mathbb{R}[x]$, for which $P(P(x))=\lfloor P^2 (x)\rfloor$ is true for $\forall x\in \mathbb{Z}$. | P(x) = 0 | 51 | 7 |
math | 8.1. Two cars are driving on a highway at a speed of 80 km/h and with an interval of 10 m. At the speed limit sign, the cars instantly reduce their speed to 60 km/h. What will be the interval between them after the speed limit sign? | 7.5\mathrm{M} | 62 | 8 |
math | 7.190. $\log _{2} 3+2 \log _{4} x=x^{\frac{\log _{9} 16}{\log _{3} x}}$. | \frac{16}{3} | 46 | 8 |
math | 7.1. In the expression $5 * 4 * 3 * 2 * 1=0$, replace the asterisks with the four arithmetic operation signs (using each sign exactly once) to make the equation true. No other signs, including parentheses, can be used. | 5-4\cdot3:2+1=0 | 58 | 12 |
math | ## Task Condition
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{(3-n)^{2}+(3+n)^{2}}{(3-n)^{2}-(3+n)^{2}}
$$ | -\infty | 56 | 3 |
math | 1. (6 points) Calculate $10.37 \times 3.4 + 1.7 \times 19.26=$ | 68 | 33 | 2 |
math | $N $ different numbers are written on blackboard and one of these numbers is equal to $0$.One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $... | N = 2 | 98 | 5 |
math | 3. If the length, width, and height of a rectangular prism are all prime numbers, and the sum of the areas of two adjacent sides is 341, then the volume of this rectangular prism $V=$ $\qquad$ . | 638 | 50 | 3 |
math | Task 1 - 270721 Jörg undertook a three-day cycling tour during his vacation. He covered half of the total planned route on the first day and one-third of the total length on the second day.
On the second day, Jörg cycled 24 km less than on the first day.
Determine the length of the route that Jörg still had left for ... | 24\mathrm{~} | 89 | 7 |
math | \section*{Exercise 3 - 171033}
Jens, Uwe, Dirk, and Peter are discussing which number set the number \(z\), defined by the term
\[
z=\frac{\lg (7-4 \sqrt{3})}{\lg (2-\sqrt{3})}
\]
belongs to.
Jens says that \(z\) is a natural number; Dirk thinks that the number \(z\) is a rational number; Uwe believes that \(z\) is... | 2 | 131 | 1 |
math | 28.62. Calculate the limit $\lim _{x \rightarrow 0} \frac{\tan x - x}{x - \sin x}$.
### 28.10. The number of roots of the equation | 2 | 49 | 1 |
math | B3. Maja designed floral arrangements. If she put 7 flowers in each vase, four flowers would remain, and if she put 8 flowers in each vase, two flowers would be missing. How many vases and how many flowers does Maja have? | 6 | 54 | 1 |
math | 3B. Find the largest divisor of 1001001001 that is less than 10000. | 9901 | 30 | 4 |
math | 1. Tyler has an infinite geometric series with sum 10 . He increases the first term of his sequence by 4 and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum 15. Compute the common ratio of Tyler's series.
Proposed by: Isabella Quan | \frac{1}{5} | 68 | 7 |
math | 4. Given four positive integers
$a, b, c, d$ satisfy:
$$
a^{2}=c(d+20), b^{2}=c(d-18) \text {. }
$$
Then the value of $d$ is $\qquad$ | 180 | 57 | 3 |
math | 2.2. A farmer wants to enclose a rectangular plot of land with an electric fence 100 m long on three sides, the plot being located next to a river, so that the fence together with a section of the riverbank as the fourth side forms a rectangle. What should the dimensions of this rectangle be so that the area of the enc... | AB=25 | 78 | 4 |
math | 19. Shan solves the simultaneous equations
$$
x y=15 \text { and }(2 x-y)^{4}=1
$$
where $x$ and $y$ are real numbers. She calculates $z$, the sum of the squares of all the $y$-values in her solutions.
What is the value of $z$ ? | 122 | 75 | 3 |
math | 7. Let $f(x)$ be a monotonic function defined on $(0,+\infty)$, for any $x>0$ we have $f(x)>-\frac{4}{x}, f\left(f(x)+\frac{4}{x}\right)=3$, then $f(8)=$ $\qquad$ . | \frac{7}{2} | 71 | 7 |
math | 1. Two-headed and seven-headed dragons came to a meeting. At the very beginning of the meeting, one of the heads of one of the seven-headed dragons counted all the other heads. There were 25 of them. How many dragons in total came to the meeting? | 8 | 56 | 1 |
math | Example 1 (Problem from the 35th IMO) Find all ordered pairs of positive integers $(m, n)$ such that $\frac{n^{3}+1}{m n-1}$ is an integer.
Find all ordered pairs of positive integers $(m, n)$ such that $\frac{n^{3}+1}{m n-1}$ is an integer. | (2,1),(3,1),(1,2),(1,3),(2,2),(5,2),(5,3),(2,5),(3,5) | 77 | 37 |
math | ## Condition of the problem
Compose the equation of the tangent to the given curve at the point with abscissa $x_{0}$.
$$
y=\frac{x^{29}+6}{x^{4}+1}, x_{0}=1
$$ | 7.5x-4 | 56 | 6 |
math | 22nd CanMO 1990 Problem 1 A competition is played amongst n > 1 players over d days. Each day one player gets a score of 1, another a score of 2, and so on up to n. At the end of the competition each player has a total score of 26. Find all possible values for (n, d). | (n,)=(3,13),(12,4),(25,2) | 79 | 18 |
math | Find all positive integers $k$ such that there exist two positive integers $m$ and $n$ satisfying
\[m(m + k) = n(n + 1).\] | \mathbb{N} \setminus \{2, 3\} | 37 | 17 |
math | Three. (50 points)
There are 12 rabbit holes, and holes $1,2,3,4,5,6,7,8,9$ contain a total of 32 rabbits. Holes $4,5,6,7,8,9,10,11,12$ contain a total of 28 rabbits, holes $1,2,3,7,8,9,10,11,12$ contain a total of 34 rabbits, and holes $1,2,3,4,5,6,10,11,12$ contain a total of 29 rabbits. How many different possible d... | 18918900 | 159 | 8 |
math | Find all $ x,y,z\in\mathbb{N}_{0}$ such that $ 7^{x} \plus{} 1 \equal{} 3^{y} \plus{} 5^{z}$.
[i]Alternative formulation:[/i] Solve the equation $ 1\plus{}7^{x}\equal{}3^{y}\plus{}5^{z}$ in nonnegative integers $ x$, $ y$, $ z$. | (0, 0, 0) | 93 | 10 |
math | Solve the following equation:
$$
1.2065^{x}+1.2065^{x+1}+1.2065^{x+2}+1.2065^{x+3}=1.2065^{10}
$$ | x\approx1 | 64 | 4 |
math | 一、Fill in the Blanks (Total 3 questions, 10 points each)
1. In a 1000-meter race, when A reaches the finish line, B is 50 meters from the finish line; when B reaches the finish line, C is 100 meters from the finish line. So when A reaches the finish line, C is $\qquad$ meters from the finish line. | 145 | 88 | 3 |
math | ## Task 2 - 330722
Determine all four-digit natural numbers $z$ that satisfy the following conditions (1) and (2)!
(1) The number $z$ is divisible by 24.
(2) The second digit of the number $z$ is a 1, and the third digit of $z$ is a 3. | 2136,5136,8136 | 81 | 14 |
math | 10.4. In an initially empty room, every minute either 2 people enter or 1 person leaves. Can there be exactly 2018 people in the room after 2019 minutes? | No | 45 | 1 |
math | Example 12 (1995 National High School League Question) Given the family of curves $2(2 \sin \theta-\cos \theta+3) x^{2}-(8 \sin \theta+$ $\cos \theta+1) y=0$ ( $\theta$ is a parameter). Find the maximum value of the length of the chord intercepted by the family of curves on the line $y=2 x$.
| 8\sqrt{5} | 91 | 6 |
math | The target below is made up of concentric circles with diameters $4$, $8$, $12$, $16$, and $20$. The area of the dark region is $n\pi$. Find $n$.
[asy]
size(150);
defaultpen(linewidth(0.8));
int i;
for(i=5;i>=1;i=i-1)
{
if (floor(i/2)==i/2)
{
filldraw(circle(origin,4*i),white);
}
else
{
filldraw(circle(origin,4*i),red... | 60 | 125 | 2 |
math | 4・184 For any three positive integers $x, y, z$, let
$$f(x, y, z)=[1+2+3+\cdots+(x+y-2)]-z$$
Find all positive integer quadruples $(a, b, c, d)$, such that
$$f(a, b, c)=f(c, d, a)=1993$$ | (23,42,23,42) | 85 | 13 |
math | 13. Fill in the following squares with $1,2,3,4,5,6,7,8,9$, using each number only once, then the largest four-digit number is ( ).
口口口口+口口口+口口=2115
Note: The squares (口) represent blank spaces to be filled with the numbers. | 1798 | 77 | 4 |
math | 3. The sum of 20 natural numbers is 2002. Find the greatest value that their GCD can take. | 91 | 28 | 2 |
math | Example 5 Find the maximum constant $k$, such that $\frac{k a b c}{a+b+c} \leqslant(a+b)^{2}+(a+b+4 c)^{2}$ holds for all positive real numbers $a, b, c$.
| 100 | 58 | 3 |
math | 15. For any positive integer $m$, the set
$$
\{m, m+1, m+2, \cdots, m+99\}
$$
has the property that in any $n(n \geqslant 3)$-element subset, there are always three elements that are pairwise coprime. Find the minimum value of $n$. | 68 | 80 | 2 |
math | 10. The sequence $a_{0}, a_{1}, \cdots, a_{n}$ satisfies
$$
a_{0}=\sqrt{3}, a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}},
$$
where, $[x]$ denotes the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$. Then $a_{2016}=$ $\qquad$ . | 3024+\sqrt{3} | 110 | 9 |
math | I2.1 It is given that $m, n>0$ and $m+n=1$. If the minimum value of $\left(1+\frac{1}{m}\right)\left(1+\frac{1}{n}\right)$ is $a$, find the value of $a$.
I2.2 If the roots of the equation $x^{2}-(10+a) x+25=0$ are the square of the roots of the equation $x^{2}+b x=5$, find the positive value of $b$. (Reference: 2001 F... | 15 | 228 | 2 |
math | 2. (7 points) It is known that $a^{2}+b=b^{2}+c=c^{2}+a$. What values can the expression $a\left(a^{2}-b^{2}\right)+b\left(b^{2}-c^{2}\right)+c\left(c^{2}-a^{2}\right)$ take?
## Answer: 0. | 0 | 83 | 1 |
math | ## T-1
Let $\mathbb{Z}$ denote the set of all integers and $\mathbb{Z}_{>0}$ denote the set of all positive integers.
(a) A function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ is called $\mathbb{Z}$-good if it satisfies $f\left(a^{2}+b\right)=f\left(b^{2}+a\right)$ for all $a, b \in \mathbb{Z}$. Determine the largest pos... | 2 | 298 | 1 |
math | Ottó decided that for every pair of numbers $(x, y)$, he would assign a number denoted by $(x \circ y)$. He wants to achieve that the following relationships hold:
a) $x \circ y=y \circ x$
b) $(x \circ y) z=x z \circ y z$
c) $(x \circ y)+z=(x+z) \circ(y+z)$.
What should Ottó assign to the pair $(1975,1976)$? | 1975.5 | 108 | 6 |
math | 6. (5 points) There is a natural number, the difference between its smallest two divisors is 4, and the difference between its largest two divisors is 308, then this natural number is $\qquad$ | 385 | 48 | 3 |
math | I4.4 If $c$ boys were all born in June 1990 and the probability that their birthdays are all different is $\frac{d}{225}$, find $d$ | 203 | 43 | 3 |
math | 1. [4] A hundred friends, including Petya and Vasya, live in several cities. Petya learned the distance from his city to the city of each of the remaining 99 friends and added these 99 numbers. Vasya did the same. Petya got 1000 km. What is the largest number Vasya could have obtained? (Consider the cities as points on... | 99000 | 117 | 5 |
math | 3. (10 points) A barrel of oil, the oil it contains is $\frac{3}{5}$ of the barrel's full capacity. After selling 18 kilograms, 60% of the original oil remains. Therefore, this barrel can hold $\qquad$ kilograms of oil. | 75 | 62 | 2 |
math | 9.4. Six different natural numbers from 6 to 11 are placed on the faces of a cube. The cube was rolled twice. The first time, the sum of the numbers on the four side faces was 36, and the second time it was 33. What number is written on the face opposite the one with the number 10? Justify your answer. | 8 | 81 | 1 |
math | (12) Let $[x]$ denote the greatest integer not exceeding $x$, $a_{k}=\left[\frac{2009}{k}\right]$, $k=1$,
$2, \cdots, 100$, then the number of different integers among these 100 integers is $\qquad$ | 69 | 74 | 2 |
math | 15. Given the parabola $y^{2}=2 p x$ passes through the fixed point $C(1,2)$, take any point $A$ on the parabola different from point $C$, the line $A C$ intersects the line $y=x+3$ at point $P$, and a line parallel to the $x$-axis through point $P$ intersects the parabola at point $B$.
(1) Prove that the line $A B$ pa... | 4\sqrt{2} | 129 | 6 |
math | For any positive integer $k$, let $f_{1}(k)$ be the square of the sum of the digits of $k$ when written in decimal notation, and for $n>1$, let $f_{n}(k)=f_{1}\left(f_{n-1}(k)\right)$. What is $f_{1992}\left(2^{1991}\right)$? | 256 | 87 | 3 |
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